Cuaderno+de+integrales

Integrales
www.fisicaeingenieria.es
En este document encontrarás toda la metodología para resolver cualquier tipo de integral definido o
indefinida
Luis Muñoz Mato

www.fisicaeingenieria.es Tabla de integrales
1
Tipos FormasFormasFormasFormas
SimpleSimpleSimpleSimple CompuestaCompuestaCompuestaCompuesta
Tipo potencial a ≠ -1
∫ +
=
+
1
1
a
x
dxx
a
a
∫ +
=′⋅
+
1
1
a
f
dxff
a
a 4 51
5
x dx x=∫ ( )( )
( )
312
302
1
2 1 1
31
x x
x x x dx
+ +
+ + + =∫
Tipo logarítmico
xLdx
x
=∫
1
fLdx
f
f
=
′
∫
3 1
3 3dx dx L x
x x
= =∫ ∫
2
3
3
3 1
8
8 3
= +
+∫
x
dx L x
x
Tipo exponencial
xx
edxe =∫
La
a
dxa
x
x
=∫
∫ ′=′⋅ edxfe f
La
a
dxfa f ′
=′⋅∫
2 1 2 1 2 11 1
2 2
x x x
e dx e dx e+ + +
= ⋅ =∫ ∫
2 1
2 1 2 11 1
3 3
2 2 ln 3
+
+ +
= ⋅ =∫ ∫
x
x x e
dx dx
Tipo seno
∫ = senxxdxcos ∫ =′⋅ senfdxffcos
1 1
s n 2 s n 2 2 cos 2
2 2
e x dx e x dx x= ⋅ = −∫ ∫
Tipo coseno
∫ −= xsenxdx cos ∫ −=′⋅ fdxfsenf cos ( ) ( ) ( )2 2
2 1 cos 1 s n 1x x x dx e x x+ ⋅ + + = + +∫
Tipo tangente
tgxxdx =∫
2
sec
( ) tgxdxxtg =+∫
2
1
tgxdx
x
=∫ 2
cos
1
tgfdxff =′⋅∫
2
sec
( ) tgfdxfftg =′⋅+∫
2
1
tgfdx
f
f
=
′
∫ 2
cos
2 2
3sec 3 sec 3tanx dx x dx x= =∫ ∫
2
2
7
7 sec 7 tan
cos
= =∫ ∫dx x dx x
x
( ) ( )2 2
5 5tan 5 1 tan 5tanx dx x dx x+ = + =∫ ∫
Tipo arco seno
arcsenxdx
x
=
−
∫ 2
1
1
a
x
arcsendx
xa
=
−
∫ 22
1
arcsenfdx
f
f
=
−
′
∫ 2
1
a
f
arcsendx
fa
f
=
−
′
∫ 22
( )
2
4 22
2 2
arcsen
1 1
x x
dx dx x
x x
= =
− −
∫ ∫
Tipo arco tangente
arctgxdx
x
=
+∫ 2
1
1
a
x
arctg
a
dx
xa
11
22
=
+∫
arctgfdx
f
f
=
+
′
∫ 2
1
a
f
arctg
a
dx
fa
f 1
22
=
+
′
∫
2 2
1 1 1 1
arctg
3 3 3 1 3
dx dx x
x x
= =
+ +∫ ∫
TTTT

2
TTTTipo potencialipo potencialipo potencialipo potencial ( )1a ≠ −
0dx C=∫ k dx kx C= +∫
1
1
a
a x
x dx
a
+
=
+∫
1
1
a
a f
f f dx
a
+
′⋅ =
+∫
Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos:
1.1.1.1. 4 51
5
x dx x=∫
2.2.2.2.
3 3
4
4
1
3 3
x x
dx x dx
x
− −
−
= = = −
−∫ ∫
3.3.3.3.
5
2 53
3 3
3
5 5
3
x
x dx x= =∫
4.4.4.4.
2
1 23
3 3
3
2 2
3
x
x dx x
−
= =∫
5.5.5.5. ( ) ( )
2 31
1 1
3
x dx x+ = +∫
6.6.6.6. ( )( )
( )
312
302
1
2 1 1
31
x x
x x x dx
+ +
+ + + =∫
7.7.7.7. 3 41
s n cos s n
4
e x x dx e x=∫
8.8.8.8. ( )
2 2 31
sec
3
tg x x dx tg x=∫
9.9.9.9. ( ) ( )3 5 3 2 41
1
4
tg x tg x dx tg x tg x dx tg x+ = + =∫ ∫
10.10.10.10. ( )3 3 2 4 31 1
cos 1 s n sin
4 3
xdx tg x tg x dx tg x e x x= + = = −∫ ∫
11.11.11.11. ( ) ( )3 2 2 31
s n 1 cos s n cos s n cos cos
3
sen xdx e x x dx e x e x dx x x= − = − = − +∫ ∫ ∫
Tipo logarítmicoTipo logarítmicoTipo logarítmicoTipo logarítmico

3
1
dx L x
x
=∫
f
dx L f
f
′
=∫
1.1.1.1.
3 1
3 3dx dx L x
x x
= =∫ ∫
2.2.2.2.
2
3
3
3 1
5
5
x
dx L x x
x x
+
= + +
+ +∫
3.3.3.3. ( )2
2 2
1 2 1
1
1 2 1 2
x x
dx dx L x
x x
= = +
+ +∫ ∫
4.4.4.4.
2 2
3
3 3
1 3 1
8
8 3 8 3
x x
dx dx L x
x x
= = +
+ +∫ ∫
5.5.5.5.
s n
cos
cos
e x
tg x dx dx Ln x
x
= = −∫ ∫
6.6.6.6.
cos
cotg s n
s n
x
x dx dx L e x
e x
= =∫ ∫
Tipo exponencialTipo exponencialTipo exponencialTipo exponencial
x x
e dx e=∫
x
x a
a dx
La
=∫
f f
e f dx e′⋅ =∫
f
f a
a f dx
La
′⋅ =∫
12.12.12.12. 2 1 2 1 2 11 1
2 2
x x x
e dx e dx e+ + +
= ⋅ =∫ ∫
13.13.13.13.
3
3
3
x
x
dx
L
=∫
14.14.14.14.
3
3 3 2
32 2
2
x
xx
x
dx dx
L
 
 
   = = 
  
 
 
∫ ∫
15.15.15.15.
2 2 21 1
2
2 2
x x x
x e dx x e dx e= ⋅ =∫ ∫
16.16.16.16. sin sin
cosx x
e x dx e=∫
17.17.17.17.
2 2
s n s n
2s n cos s ne x e x
e e x x dx e e x dx e= =∫ ∫

4
Tipo senoTipo senoTipo senoTipo seno
s n cose xdx x=−∫
s n cose f f dx f′⋅ = −∫
18.18.18.18.
1 1
s n 2 s n 2 2 cos2
2 2
e x dx e x dx x= ⋅ = −∫ ∫
19.19.19.19. ( ) ( ) ( )
1 1
s n 2 6 s n 2 6 2 cos 2 6
2 2
e x dx e x dx x+ = + ⋅ = − +∫ ∫
20.20.20.20. ( ) ( ) ( )2 2 21 1
s n 3 s n 2 cos 3
2 2
x e x dx e x x dx x⋅ + = ⋅ = − +∫ ∫
21.21.21.21. ( ) ( ) ( )2 2
2 1 s n 1 cos 1x e x x dx x x+ ⋅ + + = − + +∫
22.22.22.22.
( )
( ) (
s n 1
s n cos
e Lx
dx e Lx dx L
x x
= ⋅ = −∫ ∫
23.23.23.23. ( ) ( )s n cosx x x
e e e dx e= −∫
24.24.24.24. ( )
1 1 c
s n5 s n5 5 cos5
5 5
e x dx e x dx x= ⋅ = − =−∫ ∫
25.25.25.25. ( ) ( ) ( )
1 1
s n 7 8 s n 7 8 7 cos 7 8
7 7
e x dx e x dx x+ = + ⋅ =− +∫ ∫
Tipo cosenoTipo cosenoTipo cosenoTipo coseno
cos s nxdx e x=∫ cos s nf f dx e f′⋅ =∫
26.26.26.26.
1 1
cos2 cos2 2 sin 2
2 2
x dx x dx x= ⋅ =∫ ∫
27.27.27.27. ( ) ( ) ( )
1 1
cos 2 1 cos 2 1 2 s n 2 1
2 2
x dx x dx e x+ = + ⋅ = +∫ ∫
28.28.28.28. ( ) ( )2 21
cos 1 cos
2
x x dx x⋅ + =∫ ∫
29.29.29.29. ( ) ( ) ( )2 2
2 1 cos 1 s n 1x x x dx e x x+ ⋅ + + = + +∫
30.30.30.30.
( )
( ) ( )
cos 1
cos s n
Lx
dx Lx dx e Lx
x x
= ⋅ =∫ ∫
31.31.31.31. cos s nx x x
e e dx e e=∫
32.32.32.32. ( ) ( )2 3 3 2 3
3 cos 9 cos 9 3 s nx x dx x x dx e x+ = + ⋅ = +∫ ∫ ∫
33.33.33.33. ( ) ( ) ( )2 3 3 2 31 1
cos 1 cos 1 3 s n 1
3 3
x x dx x xdx e x+ = + ⋅ = +∫ ∫

5
Tipo tangenteTipo tangenteTipo tangenteTipo tangente
2
sec tanxdx x=∫
2
sec tanf f dx f′⋅ =∫
34.34.34.34. 2 2
3sec 3 sec 3tanx dx x dx x= =∫ ∫
35.35.35.35. 2 2
2
7
7sec 7 sec 7 tan
cos
dx x dx x dx x
x
= = =∫ ∫ ∫
36.36.36.36. ( ) ( )2 2
5 5tan 5 1 tan 5tanx dx x dx x+ = + =∫ ∫
37.37.37.37. ( ) ( ) ( )2 2 3 2 3 2 3 9
3 sec 9 sec 9 3 tanx x dx x x dx x +
+ = + ⋅ =∫ ∫
38.38.38.38. ( ) ( ) ( )2 21 1
sec 2 1 sec 2 1 2 tan 2 1
2 2
x dx x dx x+ = + ⋅ = +∫ ∫
39.39.39.39. ( ) ( )4 2 2 2 2 2 31
sec 1 tan sec sec tan sec tan tan
3
x dx x x dx x x x x x dx= + = + = +∫ ∫ ∫
40.40.40.40. ( )2 2
tan 1 tan 1 tanx dx x dx x x = + − = − ∫ ∫
Tipo cotangenteTipo cotangenteTipo cotangenteTipo cotangente
2
cosec cotgx dx x=−∫
2
cosec cotgf f dx f′⋅ =−∫
41.41.41.41. 2 2
3 cosec 3 cosec 3 cotgx dx x dx x= = −∫ ∫
42.42.42.42. 2 2
2
8
8 cosec 8 cosec 8 cotg
sin
dx x dx x dx x
x
= = = −∫ ∫ ∫
43.43.43.43. ( ) ( )2 2
5 5 cotg 5 1 cotg 5 cotgx dx x dx x+ = + = −∫ ∫
44.44.44.44. ( ) ( ) ( )2 2 1
cosec 2 1 cosec 2 1 2 cotg 2 1
2
x dx x dx x+ = + ⋅ = − +∫ ∫
45.45.45.45. ( )2 2
cotg 1 cotg 1 cotgx dx x dx x x = + − = − − ∫ ∫
46.46.46.46. ( ) ( )4 2 2 2 2 2
cosec 1 cotg cosec cosec cotg cosecx dx x x dx x x x dx= + = + =∫ ∫ ∫
31
cotg cotg
3
x x dx− −

6
Tipo arco seno (=arco coseno)Tipo arco seno (=arco coseno)Tipo arco seno (=arco coseno)Tipo arco seno (=arco coseno)
2
1
arcsen
1
=
−
∫ dx x
x 2
arcsen
1
′
=
−
∫
f
dx f
f
47.47.47.47.
( )
2
4 22
2 2
arcsen
1 1
x x
dx dx x
x x
= =
− −
∫ ∫
48.48.48.48.
( )
2 2
arcsen
1 1
x x
x
x x
e e
dx dx e
e e
= =
− −
∫ ∫
49.49.49.49.
( )
( )2 2
1
1
arcsen
1 1
xdx dx Lx
x L x Lx
= =
− −
∫ ∫
50.50.50.50.
( )
2
1 1 1
2 2arcsen
1 2
1
dx dx x
x x x
x
= ⋅ =
−
−
∫ ∫
Tipo arco tangente (=Tipo arco tangente (=Tipo arco tangente (=Tipo arco tangente (=----arco cotangente)arco cotangente)arco cotangente)arco cotangente)
2
1
arctg
1
=
+∫ dx x
x 2
arctg
1
′
=
+∫
f
dx f
f
EjeEjeEjeEjercicios resueltos:rcicios resueltos:rcicios resueltos:rcicios resueltos:
51.51.51.51. 2 2
1 1 1 1
arctg
3 3 3 1 3
dx dx x
x x
= =
+ +∫ ∫
52.52.52.52.
( )
22
1 1 3 1
arctg 3
1 9 3 31 3
dx dx x
x x
= =
+ +
∫ ∫
53.53.53.53.
( )
( )
2
3
23
3 1
arctg 2
1 2
x
dx x x
x x
+
= + +
+ + +
∫
54.54.54.54. ( )2
cos
arctg sin
1 sin
x
dx x
x
=
+∫

7
55.55.55.55.
( )
2
24 2
1 2 1
arctg
1 2 21
x x
dx dx x
x x
= =
+ +
∫ ∫
56.56.56.56.
( )
2 2
3
26 3
1 3 1
arctg
1 3 31
x x
dx dx x
x x
= =
+ +
∫ ∫
57.57.57.57.
( )
22
arctg
1 1
x x
x
x x
e e
dx dx e
e e
= =
+ +
∫ ∫
INTEGRALESINTEGRALESINTEGRALESINTEGRALES INDEFINIDASINDEFINIDASINDEFINIDASINDEFINIDAS
1.1.1.1. ( )
4 3 2
3 2 5
4
4 2 2
x x x
x x x dx C+ + − = + − +∫
2.2.2.2. 10
10
10
x
x
dx C
ln
= +∫
3.3.3.3. ( )2 2 2 2
2 2x x x x
x e dx x e x x dx x e x e dx⋅ = ⋅ − ⋅ ⋅ = ⋅ − ⋅ ⋅ = ∗∫ ∫ ∫
2
2
x x
u x du xdx
dv e dx v e
= ⇒ =
= ⇒ =
x x
u x du dx
dv e dx v e
= ⇒ =
= ⇒ =
( ) 2 2
2 2x x x x x x
x e x e e dx x e x e e C   ∗ = ⋅ − ⋅ − = ⋅ − ⋅ − + =  ∫
( )2 2
2 2 2 2x x x x
x e xe e C e x x C= ⋅ − + + = − + +
4.4.4.4. ( )
( )
2
1
1
2
x
x x
e
e e dx C
+
+ = +∫
5.5.5.5. ( )
1 1
3 3 3
2 23
1 1 3 1 3
4
4
x x
e x dx e dx x dx x dx dx dx
x x x xx
−− − 
+ − − + = + − − + = 
 
∫ ∫ ∫ ∫ ∫
( )
1 1
3 1 23
1
4 4 3
1 41
3
x x
e x dx ln x x dx
+
−− −
= + − ⋅ ⋅ − + =
+
∫ ∫
( )
14 1 2 133 41
3
4 14 2 11
3 3
x xx x
e ln x C
− + − +
−
= + − ⋅ − + ⋅ + =
− − ++
( )
4 2
3 3
3 3 3
4
4 8
x
e x x ln x C
x
−
− + ⋅ − − − +

8
6.6.6.6.
( )
( ) ( )
( )
4 1
4 4
4
2 11 1 1
2 1 2 1 2
2 2 4 12 1
x
dx x dx x dx C
x
− +
− − +
= + = + ⋅ ⋅ = ⋅ +
− ++
∫ ∫ ∫
( )
( )
3
3
1 1
2 1
6 6 2 1
x C C
x
−
= − ⋅ + + = − +
+
7.7.7.7.
2
3
3
7 5 5
4 7
5 3
4 7
3
x
dx ln x x C
x x
+  
= + + + 
   + + 
 
∫
8.8.8.8.
( )
( ) ( )
( )
( )
4 12
42
4 32 2
2 1 1
2 1
4 1 3
x xx
dx x x x dx C C
x x x x
+
++
= + ⋅ + = + = − +
− ++ +
∫ ∫
9.9.9.9. ( )2
x
dx x tgx tgx dx x tgx ln cosx C x tgx ln cosx C
cos x
= ⋅ − ⋅ = ⋅ − − + = ⋅ + +∫ ∫
2
1
u x du dx
v tgx dv dx
cos x
= ⇒ =
= ⇒ =
10.10.10.10. 1 1 1 1
1 1x x
x
x
dx dx dt
e e te t
e t
−
= = ⋅ =
+ + +
∫ ∫ ∫
1
x
e t x lnt
dx
tdt
= ⇒ =
=
1
x
lne lnt
xlne lnt lne
x lnt
=
= ⇒ =
=
2 2
1 1 1
1 1
x
dt dt arctgt C arctge C
t t t
t
⋅ = = + = +
+ +∫ ∫
11.11.11.11. ( ) ( )2 3 2 2 3 2 31 1
1 1 3
3 3
tg x x dx tg x x dx tgx C+ ⋅ ⋅ = + ⋅ ⋅ = +∫ ∫
12.12.12.12. ( )2 1 2 1 1 2
1 2
2 2 2 2
cos x sen x
sen xdx dx cos x dx x C
⋅  
= = − = − + 
 
∫ ∫ ∫
2 2
2 2
2
1
cos x cos x sen x
cos x sen x
− = − +
= +
1 2
2 2 2
2 2
x senx cosx
C
x senx cosx
C
⋅ ⋅
= − +
⋅
= − +

9
2
2
1 2 2
1 2
2
cos x sen x
cos x
sen x
− =
−
=
13.13.13.13. ( ) ( )2 2 2
1 1 1tg xdx tg x dx tg x dx dx tgx x C= + − = + − = − +∫ ∫ ∫ ∫
14.14.14.14. ( ) ( ) ( )2 2 2
2 1 1 1tg x dx tg x dx dx tg x dx x tgx C+ = + + = + + = + +∫ ∫ ∫ ∫
15.15.15.15.
2 2 2
1 1
2 2 2 2
x x x
x lnx dx lnx dx lnx xdx
x
⋅ ⋅ = ⋅ − ⋅ = ⋅ − =∫ ∫ ∫
2 2 2 2
1
2 2 2 2 4
x x x x
lnx C lnx C= − ⋅ + = − +
2
1
2
u lnx du dx
x
x
dv xdx v
= ⇒ =
= ⇒ =
16.16.16.16. ( )
( )
1 12 2
1
2 2 2
11 1
1 1 2
12 2 1
2
x
x x dx x x dx C
+
+
+ ⋅ ⋅ = + ⋅ ⋅ = + =
+
∫ ∫
( ) ( )
3 3
2 22 2
1 11
32 3
2
x x
C C
+ +
= + = +
17.17.17.17. ( )
( )
1 1
21
2
1 1
2
13
3 1 3
1
senxcosx
dx senx cosx dx C
senx
− +
−
− +
 +
 = + ⋅ ⋅ = + =
 +
 
∫ ∫
( )
1
2
1
3 6 1
1
2
senx
C senx C
+
= + = + +
18.18.18.18. ( )
4 4 3
31 1
1
3
x x x x x
x x x
x x x x
e e e e e
dx dx e e dx x e C
e e e e
− − + +
= + + = + + = + − + 
 
∫ ∫ ∫
19.19.19.19. 2 2 2
1 1
1 13 3
3 39 9
1
9 9 3
x
dx dx dx arcsen C
x x x
 
= = = + 
 −  − − 
 
∫ ∫ ∫

10
20.20.20.20. ( )
3
2 23
2 2 2 2 2
2
1
1 1 2 1
31 2
xx
dx x x x x dx x x C
x
+
= ⋅ + − + ⋅ ⋅ = ⋅ + − + =
+
∫ ∫
21.21.21.21.
2
2
2
2
1
1
u x du xdx
x
dv dx v x
x
= ⇒ =
= ⇒ = +
+
22.22.22.22. 2 2
1 1 1 1 1 1 1lnx
dx lnx dx lnx dx lnx C
x x x x x x x x
 
= − − − = − − − = − − + 
 
∫ ∫ ∫
2
1
1 1
u lnx du dx
x
dv dx v
x x
= ⇒ =
= ⇒ = −
23.23.23.23.
( ) ( ) ( ) ( )
2 2
2 22 3 2 3
5 5 1
5 5
4 4
x x
dx dx
cos x cos xsen x sen x
 
 + = + =
 
 
∫ ∫ ∫
( )
( ) ( )
2
3
22 3
5 3 5 4 5 5
4
3 4 4 3 4
x
dx dx cotg x tg x C
cos xsen x
−
= − + = − + +∫ ∫
24.24.24.24. ( )
( )
( )
( ) ( )2 5 2 5 2 5
2 5 2 5
1 1 1 1
2
2 2 2
x x x
x x
dx e dx e dx e C C
e e
− + − + − +
+ +
= = − =− + = − +
−∫ ∫ ∫
( )
( )
( ) ( )
2 2 2 2
2 2 2
2 2 2
2
2
1 1 1
3
2
1 1
3
2 2 1
1
3 3
x x x x C
x x x C
x x x
x C C
= ⋅ + − + ⋅ + + =
 
= − + + + =  
− − +
= + + = +

11
25.25.25.25. 2 2 22
1 1
1 1 1 1 14 22
44 4 4 2 2
1 1
4 4 2 2
x
dx dx dx dx arctg C
xx x x
 
= = = ⋅ = + 
+     + + +   
   
∫ ∫ ∫ ∫
26.26.26.26.
2 2 2
3
3 3 3
2 2 3 2
2 4
4 4 3 4 3
x x x
dx dx dx ln x C
x x x
= = = + +
+ + +∫ ∫ ∫
27.27.27.27.
( )
2
2
24 2
1 1 1 1 1 1
1 2 2 1 2 21
x
x
x
e
dx dt dt arctg t C arctg e C
e tt
= ⋅ = = + = +
+ ++
∫ ∫ ∫
28.28.28.28. ( )
2 2
3 3
6x x
e x dx e C− −
⋅ − = +∫
29.29.29.29.
2
2 2
1
2
tgx tg x
dx tgx dx C
cos x cos x
= ⋅ = +∫ ∫
Otra forma de hacerla:
( ) ( )
( )
2
3
3 2
1
2 2
cosxsenx
dx cosx senx dx C C
cos x cos x
−
−
= − − = − + = +
−∫ ∫
30.30.30.30. tgx dx ln cosx C⋅ = − +∫
31.31.31.31.
( )
( ) ( )
( )
7 3 4 7 3 4
1 1
7 7 3 4 4
7 4
1 3 7 3 4
7 4
7 4 7 4
cos x sen x dx cos x dx sen x dx
cos x dx sen x dx
sen x cos x
sen x cos x C C
− = ⋅ − ⋅ =
= ⋅ ⋅ − ⋅ = ⋅ ⋅ =
= − − + = + +
∫ ∫ ∫
∫ ∫
32.32.32.32.
( ) ( )
( )2
22 2 2
1 2 1
4
2 21 4 1 4
x x
dx dx arctg x C
x x
= = + +
+ + + +
∫ ∫
33.33.33.33. ( )
2
1
2
lnxlnx
dx lnx dx C
x x
= ⋅ = +∫ ∫
34.34.34.34. x x x x x x
e cosx dx e cosx e senx dx e cosx e senx e cosx dx ⋅ ⋅ = ⋅ + ⋅ ⋅ = ⋅ + ⋅ − ⋅ ⋅
 ∫ ∫ ∫
x x
u cosx du senxdx
dv e dx v e
= ⇒ = −
= ⇒ = x x
u senx du cosxdx
dv e dx v e
= ⇒ =
= ⇒ =

12
Resumiendo: 2
2
x x x x
x x x
x x
x
e cosx dx e cosx e senx e cosx dx
e cosx dx e cosx e senx
e cosx e senx
e cosx dx C
⋅ = + − ⋅ =
⋅ = + =
+
⋅ = +
∫ ∫
∫
∫
35.35.35.35. ( )4 2 2 2 2
1sen x dx sen x sen x dx sen x cos x dx⋅ = ⋅ ⋅ = − =∫ ∫ ∫
( )
2
2 2 2 2 2
4
sen x
sen x sen x cos x dx sen x dx
 
− ⋅ = − = 
 
∫ ∫
2 2
2 2
2
2
2
1
1 2 2
1 2
2
cos x cos x sen x
cos x sen x
cos x sen x
cos x
sen x
− = − +
= +
− =
−
=
1 4
1 2 2
2 4
1 2 1 4
2 2 8 8
cos x
cos x
dx
cos x cos x
dx
− 
 −
= − = 
 
 
 
= − − + = 
 
∫
∫
2 4
3 2 4 3 2 4
8 2 8 8 2 8
sen x sen x
cos x cos x
dx x C
 
− + = − + + 
− 
∫
3 2 4
8 4 32
sen x sen x
x C= − + +
36.36.36.36. senx senx
e cosx dx e C⋅ ⋅ = +∫
37.37.37.37. ( )3 3 3 2 3 2
1sen x cos x dx sen x cos x cosx dx sen x sen x cosx dx⋅ ⋅ = ⋅ ⋅ ⋅ = − ⋅ ⋅ =∫ ∫ ∫
( )
4 6
3 5
4 6
sen x sen x
sen x cosx sen x cosx dx C= ⋅ − ⋅ = − +∫
Otra forma de hacerlo: ( )2 3 2 3
1senx sen x cos x dx senx cos x cos x dx⋅ ⋅ ⋅ = − ⋅ ⋅ =∫ ∫
( ) ( )3 3 3 5
4 6
4 6
cos x senx dx cos x senx dx cos x senx dx cos x senx dx
cos x cos x
C
⋅ ⋅ − ⋅ ⋅ = − − + − =
= − + +
∫ ∫ ∫ ∫
38.38.38.38. ( ) ( ) ( ) ( )2 2 21 1
1 1 2 1
2 2
x cos x dx cos x x dx sen x C   ⋅ + = − + ⋅ − ⋅ = − + +   ∫ ∫

13
39.39.39.39.
( ) ( ) 22
1 1 1
2 2 2 2
111
dx t dt dt arctgt C arctg x C
tt tx x
= ⋅ ⋅ = = + = +
+++
∫ ∫ ∫
2
2
x t x t
dx tdt
= ⇒ =
=
40.40.40.40. 2
1 4
5 1 43 3 3 3
9 3 3 3 3
x
dx dx ln x ln x C
x x x
 −−  = + = − − + + +
− − + 
 
∫ ∫
1 0 9
3
3 9
1 3 0
3
3
1 0
−
−
( ) ( )
( )( )
( ) ( )
2
2
5
9 3 3
3 35
9 3 3
5 3 3
13 2 6
3
43 8 6
3
x A B
x x x
A x B xx
x x x
x A x B x
x A A
x B B
−
= +
− − +
+ + −−
=
− − +
− = + + −
= ⇒ − = ⇒ = −
= − ⇒ − = − ⇒ =
41.41.41.41. 3
3 3 5
5 5
x x
x
x x
e e
dx dx ln e C
e e
= = + +
+ +∫ ∫
42.42.42.42. ( )
( )
( )3 3 4 36 3
5 5
6 4 24 223
1 1
6 6 6
11
t t t tx x t t
dx t dt t dt dt
t t tt tx x
− −− −
= ⋅ = ⋅ ⋅ = =
− −−−
∫ ∫ ∫ ∫
6 6
5
6
x t t x
dx t dt
= ⇒ =
=
( )( )
( )( )
4 2 6 5 4
6 3 2
6 4 3 2
4 3 26 6 6 6 6
1 1
6 6
1 1 1
1
6 1
1
3
2 3 6 6 1
2
3
2 3 6 6 1
2
t t t t t t t
dt dt
t t t
t t t t dt
t
t t t t t ln t C
x x x x x ln x C
− + + +
= = =
− + +
 
= + − + − + = 
+ 
= + − + − + + + =
= + − + − + + +
∫ ∫
∫

14
1 0 0 1
1
1 1 1
0
−
1 1 1
( )( )2 2
1 1 1t t t t− = − + +
1 1 1 0 0 0 0
1
1 0 1 1 1 1
1
−
− − − −
1 0 1 −1 −1 −1
43.43.43.43.
( )
2 2
2
1 1 1 1 1
11 1 1 1
x x x x x
dx dx dx dx
xx x x x
+ + ⋅ − − −
= = = =
−− − ⋅ − −
∫ ∫ ∫ ∫
2 2
1 1
x sent t arcsenx
dx cost
cost sen t x
= ⇒ =
=
= − = −
2 2
1
1 1 1
sen t cos t cost
cost dt cost dt cost dt
sent sent sent
−
= ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ =
− − −∫ ∫ ∫
( )( )2 2
1 11
1 1 1
sent sentcos t sen t
cost dt dt dt
sent sent sent
− +−
⋅ ⋅ = = =
− − −∫ ∫ ∫
( ) 2
1 1sent dt t cost C arcsenx x C+ = − + = − − +∫
44.44.44.44. ( )
( )( ) 2 2
1 11 1 1
1 1 1 1
senx senx senx
dx dx dx dx
senx senx senx sen x cos x
⋅ − + +
= = = =
− − + −∫ ∫ ∫ ∫
2
2 2 2
1 1senx
dx dx cos x senx dx
cos x cos x cos x
− 
+ = + ⋅ ⋅ = 
 
∫ ∫ ∫
( ) ( )
( )
1
2
2
1 1
1
cosx
dx cosx senx dx tgx C tgx C
cos x cosx
−
−
= − ⋅ − = − + = + +
−∫ ∫
45.45.45.45.
( )
2 3 2
2
2 2 2
1 1
x t t t
dx t dt dt dt
t t t t tx x
= ⋅ ⋅ = = =
+ + ++
∫ ∫ ∫ ∫
2
2
x t x t
dx tdt
= ⇒ =
=
1 0 0
1
1 1
1
−
−
1 −1
Cociente t=1

15
2
1
2 1 2 1
1 2
t
t dt t ln t C
t
  
= − + = − + + + =  
+   
∫
2
2 2 1 2 2 1t t ln t C x x ln t C= − + + + = − + + +
46.46.46.46.
2 3
2
2 2
1
3
tg x tg x
dx tg x dx C
cos x cos x
= ⋅ ⋅ = +∫ ∫
47.47.47.47. 2 2
1tgx
tgx tgxe
dx e dx e C
cos x cos x
= ⋅ = +∫ ∫
48.48.48.48. 2
2 2 2
5 1 5 2 1 10 1
9 5
9 5 2 9 5 2 9 5 2
x x x
dx dx dx ln x C
x x x
⋅
= = = + +
+ + +∫ ∫ ∫
49.49.49.49. ( )
3 2 3
28 8 1 8
8 1
1 3
x x x x
dx x dx x C
x
+ + +
= + = + +
+∫ ∫
8 8 1 1
1
8 0 1
0
−
− −
8 0 1
50.50.50.50. ( )2 2 22 2 2 2
1 2sen x cos x sen xsen x sen x sen x cos x
dx dx dx
senx cosx senx cosx senx cosx
+ ++ + +
= = =
⋅ ⋅ ⋅∫ ∫ ∫
2 2
2
2
sen x cos x senx cosx
dx dx dx dx
senx cosx senx cosx cosx senx
+ = + =
⋅ ⋅∫ ∫ ∫ ∫
2ln cosx ln senx C− + +
51.51.51.51. ( )
( )
4
5
5 4
1
4 4
senxcosx
dx senx cosx dx C C
sen x sen x
−
−
= ⋅ ⋅ = + =− +
−∫ ∫
52.52.52.52. ( ) ( )
( ) ( )
1 112 22 2
1
2 2
2
1 12
1 2
1 111 2 2
x xx
dx x x dx C C
x
− +
− − −
= − − − = − + = − + =
− +−
∫ ∫
2
2 1 x C= − − +
53.53.53.53.
( )
2
4 22
2 1
2
1 1
x
dx x dx arcsenx C
x x
= ⋅ ⋅ = +
− −
∫ ∫
54.54.54.54. 2 1 2 2
2 2 4
cos x x sen x
cos x dx dx C
+
⋅ = = + +∫ ∫

16
2 2
2 2
2
1
cos x cos x sen x
cos x sen x
= −
= +
2
2
1 2 2
1 2
2
cos x cos x
cos x
cos x
+ =
+
=
55.55.55.55. x cosx dx xsenx senx dx xsenx cosx C⋅ ⋅ = − ⋅ = + +∫ ∫
u x du dx
dv cosx dx v senx
= ⇒ =
= ⋅ ⇒ =
56.56.56.56. 2
2
1
1
1
arcsenx dx x arcsenx x dx x arcsenx x C
x
⋅ = ⋅ − ⋅ = ⋅ + + +
−
∫ ∫
2
1
1
u arcsenx du dx
x
dv dx v x
= ⇒ =
−
= ⇒ =
57.57.57.57.
3
2 2
4 4
1 1
x x
dx x dx
x x
− − 
= + = 
− − 
∫ ∫
23 5
3 52 2 1 1
1 1 2 2 2
x
x dx ln x ln x C
x x
− 
 = + + = − − + + +
− + 
 
∫
( )( )2
1 1 1x x x− = − +
3 2
3
4 1
4
x x
x x x
x
− −
− +
−
3
2 2
4 4
1 1
x x
x
x x
− −
= +
− −
( ) ( )
( )( )2 2
1 14 4
1 1 1 1 1 1
A X B Xx A B x
x X X x X X
+ + −− −
= + ⇒ =
− − + − − +
( ) ( )4 1 1X A X B X− = + + −
31 3 2
2
51 5 2
2
X a A
X B B
= ⇒ − = ⇒ = −
= − ⇒ − = − ⇒ =
58.58.58.58.
2 2
2
2
1 1
1 cos x cos xdx dx dx ln tgx C
senx cosxsenx cosx tg x
cos x
= = = +
⋅⋅∫ ∫ ∫
59.59.59.59. ( ) ( )2 23 1
1 1 1
senx sen x senx cos xsen x
dx dx dx
cosx cosx cosx
−
= = =
− − −∫ ∫ ∫
( )( )
( )
1 1
1
1
senx cosx cosx
dx senx cosx dx
cosx
− +
= + =
−∫ ∫
2
2
sen x
senx dx senx cosx dx cosx C= ⋅ + ⋅ ⋅ = − + +∫ ∫

17
60.60.60.60.
2 3
2 1
2 2 2 1
1 1 11
x t t
dx t dt dt t t dt
t t tx
 
= ⋅ ⋅ = = − + − = 
+ + ++  
∫ ∫ ∫ ∫
2
2x t x t dx tdt= ⇒ = ⇒ =
1 0 0 0
1
1 1 1
1
−
− −
1 −1 1 −
3
2 1
1
1 1
t
t t
t t
= − + −
+ +
3 2
3 2
3
2 1
3 2
2
2 2 1
3
2
2 2 1
3
t t
t ln t C
t t t ln t C
x x x ln x C
 
= − + − + + = 
 
= − + − + + =
= − + − + +
61.61.61.61.
( ) ( )
2 2
2 1 1 1
2 2 2
21 4 1 2 1 2
x
x x
x x x
dx dx ln dx
ln
= ⋅ ⋅ = ⋅ ⋅ ⋅ =
− − −
∫ ∫ ∫
( )1
2
2
x
arcsen C
ln
= ⋅ +
2 2 2 2
2
x x x dt
t ln dx dt dx
ln
= ⇒ ⋅ = ⇒ =
62.62.62.62.
( ) ( ) ( )2
1 1 1 1
3 33x
dx dt dt
t t t t te
− = ⋅ = =
− −−∫ ∫ ∫
1x
e t x lnt dx dt
t
= ⇒ = ⇒ =
( ) ( )
( ) ( )
( )
2
2 2 2 2
3 31 1
3 3 3 3
At t B t CtA B C
t t t t t t t t t
− + − +
= + + ⇒ =
− − − −
( ) ( ) 2
1 3 3At t B t Ct= − + − +
10 1 3
3
13 1 9
9
t B B
t C C
= ⇒ = − ⇒ = −
= ⇒ = ⇒ =
2
1 1 1
1 1 19 3 9 3
3 9 3 9
dt ln t ln t C
t t t t
 − −
 = + + = − + + − + =
− 
 
∫

18
1 1 1
3
9 3 9
x
x
x ln e C
e
= − + + − +
63.63.63.63.
( ) ( )
( ) 22
1 1 1
2 2
12 1 2 1
dx t dt dt
tx x t t
= ⋅ − = − =
+ − − − − 
∫ ∫ ∫
2 2
1 1 1 2x t x t x t dx tdt− = ⇒ − = ⇒ = − ⇒ = −
2 2 1arctgt C arctg x C= − + = − − +
64.64.64.64. ( )
( )
( )224
3 2
24
1 1
4 4 4
1 1
t t t tx x t t
dx t dt t dt dt
t t t t tx x
+ ++ +
= = = =
− − −−∫ ∫ ∫ ∫
4 24
4x t x t dx t dt= ⇒ = ⇒ =
1 1 0 0 0
1 2 2 2
2 Resto1 2 2 2 =
3 2
2 2 2t t t⇒ + + +
4 3 4 3
3 2 22 2
4 4 2 2 2 4 2 2 1
1 1 4 3
t t t t
dt t t t dt t t ln t C
t t
 +  
= = + + + + = + + + + − +  
− −   
∫ ∫
4 3 2 34 4 48 8
4 8 8 1 4 8 8 1
3 3
t t t t ln t C x x x x ln x C= + + + + − + = + + + + − +
65.65.65.65.
( )
3 2
3
4 334
1 1
4 4 4
11
t t
dx t dt dt dt
t t tt tx x
= ⋅ ⋅ = = =
− −−−
∫ ∫ ∫ ∫
4 3
4x t x t dx t dt= ⇒ = ⇒ =
2
3 34
3
4 3 4 4
1 1
3 1 3 3
t
dt ln t C ln x C
t
= = − + = − +
−∫
66.66.66.66.
4
2 3
2 2 2
3 48 48
3 12 12
4 4 4
x
dx x dx x x dx
x x x
 
= + + = + + = 
− − − 
∫ ∫ ∫

19
4
4 2
2
2
3
3 12
12
48
x
x x
x
x
− +
−12 +
48
( ) ( )
( )( )
( ) ( )
2
2
48
4 2 2
2 248
4 2 2
48 2 2
2 48 4 12
2 48 4 12
A B
x x x
A x B x
x x x
A x B x
x A A
x B B
= +
− − +
+ + −
=
− − +
= + + −
= ⇒ = ⇒ =
= − ⇒ = − ⇒ = −
3
3
92 92
12
2 2
12 12 2 12 2
x x dx
x x
x x ln x ln x C
 
= + + − = 
− + 
= + + − − + +
∫
67.67.67.67.
( )
1 1 1 1 1 2
1 1 1 1
x
x
e t t
dx dt dt dt
e t t t t t t
+ + +  
= ⋅ = = + = 
− − − − 
∫ ∫ ∫ ∫
1x
e t x lnt dx dt
t
= ⇒ = ⇒ =
( ) ( )
( )
( )
11 1
1 1 1 1
A t Btt A B t
t t t t t t t t
− ++ +
= + ⇒ =
− − − −
( )
0 1
1 1
1 2
t A
t A t Bt
t B
= ⇒ =
+ = − + ⇒
= ⇒ =
2 1 2 1 2 1x x x
ln t ln t C lne ln e C x ln e C= + − + = + − + = + − +
68.68.68.68. 1 1 1 1
2 2 2
1 1 11 1
t t
dx t dt dt dt
t t tx
+ −
= ⋅ ⋅ = = =
+ + ++ +
∫ ∫ ∫ ∫
1 1 1
2 2
1 1 1
t
dt dt dt dt
t t t
+   
= − = − =   + + +   
∫ ∫ ∫ ∫
2 2
1 1 1 2x t x t x t dx tdt+ = ⇒ + = ⇒ = − ⇒ =
69.69.69.69.
( )2 2
2 1 3 1 3 1
3 3 1
1 1
x
dx dx ln x ln x C
x x x x x x
+ − − 
= + + = − + + − + 
− − 
∫ ∫
( ) ( )
( ) ( )
( )
2
2 2 2 2
1 12 1 2 1
1 1 1 1
Ax x B x Cxx A B C x
x x x x x x x x x
− + − ++ +
= + + ⇒ = =
− − − −

20
( ) ( ) 2
2 1 1 1
0 1 1
1 3 3
2 5 2 4 5 2 1 12 6 2 3
x Ax x B x Cx
x B B
x C C
x A B C A A A
+ = − + − +
= ⇒ = − ⇒ = −
= ⇒ = ⇒ =
= ⇒ = + + ⇒ = − + ⇒ − = ⇒ = −
70.70.70.70. 2 2
1 1 1
2 2 2
1 1 1
dx t dt dt arcsent C
x x t t t
= ⋅ ⋅ = = + =
− − −
∫ ∫ ∫
2arcsen x C= +
2
2x t x t dx tdt= ⇒ = ⇒ =
71.71.71.71. ( )3 2 2
1cos x dx cos x cosx dx sen x cosx dx⋅ = ⋅ ⋅ = − ⋅ ⋅ =∫ ∫ ∫
( )
3
2
3
sen x
cosx sen x cosx dx senx C= − ⋅ = − +∫
72.72.72.72.
3
3 3 3 33
3 3 3 3
sen x
xcos x cos x xcos x
x sen x dx dx C
−
⋅ ⋅ = + = − + + =∫ ∫
3
3
3
u x du dx
cos x
dv sen x dx v
= ⇒ =
= ⋅ ⇒ = −
3 3
3 9
xcos x sen x
C= − + +
73.73.73.73. 2 2
1 1 2
1 2 1
x
arctgx dx x arctgx x x x arctgx dx
x x
⋅ = ⋅ − ⋅ = ⋅ − =
+ +∫ ∫ ∫
2
1
1
u arctgx du dx
x
dv dx v x
= ⇒ =
+
= ⇒ =
21
1
2
x arctgx ln x C= ⋅ − + +
74.74.74.74. 2 2 2
1 1x dx sen t cost dt cos t cost dt cost cost dt− = − ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ =∫ ∫ ∫ ∫
2 2
1 1
x sent t arcsenx
dx costdt cost sen t x
= ⇒ =
= ⇒ = − = −
2 2
2 2
2
1
cos t cos t sen t
cos t sen t
= −
= +
2
2
1 2 2
1 2
2
cos t cos t
cos t
cos t
+ =
+
=

21
2
2 2
1 2 2 2
2 2 2 2 2
sen t sent cost
cos t t t
cos t dt dt C C
⋅
+
= ⋅ = = + + = + + =∫ ∫
2
1
2 2 2 2
t sent cost arcsenx x x
C C
⋅ ⋅ −
= + + = + +
75.75.75.75. ( )2 2 2 2 2 2 2
1a x dx a a sen t a cost dt a sen t a cost dt− = − ⋅ ⋅ ⋅ = − ⋅ ⋅ ⋅ =∫ ∫ ∫
x
x a sent t arcsen dx a cost dt
a
= ⋅ ⇒ = ⇒ = ⋅ ⋅
2 2
a cos t a cost dt a cost a cost dt⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ =∫ ∫
2 2 2 21 2 2
2 2 4
cos t t sen t
a cos t dt a dt a C
+  
= ⋅ = = + + 
 
∫ ∫
2
2 2
1
2 2 2 2
x xx
arcsen
a at sent cost aa C a C
   − 
⋅    
= + + = + + =      
 
 
2
2 2
2 2
x
a arcsen
x a xa C
⋅
⋅ −
= + +
76.76.76.76. ( ) ( )
2 3
2 2
2 2
2 3
x
x x x xe x
e x dx e x xe dx x e dx+ = + + = + + ⋅ ⋅ =∫ ∫ ∫
x x
u x du dx
dv e dx v e
= ⇒ =
= ⇒ =
( )
2 3 2 3
2 2 2
2 3 2 3
x x
x x x xe x e x
x e e dx xe e C= + + ⋅ − = + + − +∫
77.77.77.77. ( )4 2 2 2 2
1sen x dx sen x sen x dx sen x cos x dx⋅ = ⋅ ⋅ = − =∫ ∫ ∫
( )2 2 2 2 2
4
sen x
sen x sen xcos x dx sen x dx
 
− = − = 
 
∫ ∫
2 2
2 2
2
2
1
1 2 2
cos x cos x sen x
cos x sen x
cos x sen x
− = +
= +
− =
21 2
2
cos x
sen x
−
=

22
1 2 1 4 2 4
2 8 2 4 8 32
cos x cos x x sen x x sen x
dx C
− − 
= − = − − + + 
 
∫
3 2 4
8 4 32
x sen x sen x
C= − + +
78.78.78.78. 2 2 2
2 2ln x dx x ln x lnx dx xln x x lnx dx ⋅ = ⋅ − ⋅ = − ⋅ − =
 ∫ ∫ ∫
2 1
2u ln x du lnx dx
x
dv dx v x
= ⇒ = ⋅ ⋅
= ⇒ =
1
u lnx du dx
x
dv dx v x
= ⇒ =
= ⇒ =
2
2 2x ln x lnx x C= ⋅ − + +
79.79.79.79. 3 3 2
3x senx dx x cosx x cosx dx⋅ ⋅ = − ⋅ + ⋅ ⋅ =∫ ∫
3 2
3u x du x dx
dv senx dx v cosx
= ⇒ =
= ⋅ ⇒ = −
2
2u x du xdx
dv cosx dx v senx
= ⇒ =
= ⋅ ⇒ =
u x dv dx
du senxdx v cosx
= ⇒ =
= ⇒ = −
3 2
3 2x cosx x senx x senx dx = − ⋅ + − ⋅ ⋅ = ∫
3 2
3 6x cosx x senx x cosx cosx dx − ⋅ + ⋅ − − ⋅ + ⋅ =
 ∫
3 2
3 6 6x cosx x senx x cosx senx C= − ⋅ + ⋅ + ⋅ − +
80.80.80.80. 2 2 2 1 2
2
2 2 2 2 2
x x x x
x x x
x dx dx C
ln ln ln ln ln
− − − −
− ⋅ ⋅ −
⋅ = − + = − + ⋅ + =∫ ∫
2
2
2
x
x
u x du dx
dv dx v
ln
−
−
= ⇒ =
= ⋅ ⇒ = −
( )
2
2 2
2 2
x x
x
C
ln ln
− −
⋅
= − +
81.81.81.81. 3 3 3 3 3 3 3 3 3x x x x x x
cosx dx senx ln senx dx senx ln cosx ln cosx d⋅ ⋅ = ⋅ − ⋅ ⋅ = ⋅ − − ⋅ + ⋅ ⋅
∫ ∫ ∫
3 3 3x x
u du ln dx
dv cosx dx v senx
= ⇒ = ⋅ ⋅
= ⋅ ⇒ =
3 3 3x x
u du ln dx
dv senx dx v cosx
= ⇒ = ⋅
= ⋅ ⇒ = −
( )
2
3 3 3 3 3 3x x x x
cosx dx senx ln cosx ln cosx dx⋅ = + ⋅ ⋅ − ⋅ ⋅ =∫ ∫
( )
2
1 3 3 3 3 3x x x
ln cosx dx senx ln cosx + ⋅ ⋅ = + ⋅ =
 ∫

23
( )
2
3 3 3
3
1 3
x x
x senx ln cosx
cosx dx C
ln
+ ⋅
⋅ ⋅ = +
+
∫
82.82.82.82.
3 3 3
2 21 1
3 3 3 3
x x x
x lnx dx lnx dx lnx x dx
x
⋅ = ⋅ − ⋅ ⋅ = − =∫ ∫ ∫
3
2
1
3
u lnx du dx
x
x
dv x dx v
= ⇒ =
= ⇒ =
3 3 3 3
1
3 3 3 3 9
x x x x
lnx C lnx C= − ⋅ + = − +
83.83.83.83. 2 2
1 1 1
2 2
2
dx dx tg x C
x cos x cos x x
= ⋅ ⋅ = +
⋅
∫ ∫
84.84.84.84. ( )2 2
1 1 1
1 1
dx dx arcsen lnx C
xx ln x ln x
= ⋅ ⋅ = +
− −
∫ ∫
85.85.85.85.
( ) ( )
2 2
1
7 1 1 27 7 7
2 49 7 72 7
1
2
x
x x
x x
dx dx ln dx
ln x
= ⋅ ⋅ = ⋅ ⋅ ⋅ =
+ +
+
∫ ∫ ∫
2 2
1 1 1 1 1 1 7 7
7 7 2
7 2 7 2 27 7
1 1
2 2
x
x
x x
ln
ln dx dx
ln ln
⋅
= ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ =
   
+ +   
   
∫ ∫
2 7 2 2 7
2 7 2 7 22
x x
arctg C arctg C
ln ln
   ⋅
= ⋅ + = ⋅ +  
   
86.86.86.86.
2
3 2
1
1
2 5 1 2 1 12 2 2 1 2
2 1 2 2 2
x x
dx dx ln x ln x ln x C
x x x x x x
 
− + −
= + + = + − − + + 
+ − − + 
 
∫ ∫

24
1 1 2 0
0
0 0 0
1 1 2 0
1
1 2
1 2 0
2
2
0
−
−
−
−
1
( )( )
( )( ) ( ) ( )
( )( )
3 2
2
3 2
2
3 2
2 1 2
2 5 1
2 1 2
1 2 2 12 5 1
2 1 2
x x x x x x
x x A B C
x x x x x x
A x x Bx x Cx xx x
x x x x x x
+ − = − +
+ −
= + +
+ − − +
− + + + + −+ −
=
+ − − +
( )( ) ( ) ( )2
2 5 1 1 2 2 1x x A x x Bx x Cx x+ − = − + + + + −
10 1 2
2
1 6 3 2
12 3 6
2
x A A
x B B
x C C
= ⇒ − = − ⇒ =
= ⇒ = ⇒ =
= − ⇒ − = ⇒ = −
87.87.87.87.
2 2
1 1
1 1
1 1 1 1
x
x
e t t
dx dt dt dt ln t C
e t t t t
− 
= ⋅ = = + = − + + 
+ + + + 
∫ ∫ ∫ ∫
1x
e t x lnt dx dt
t
= ⇒ = ⇒ = 1x x
e ln e C= − + +
1 0
1
1
1
−
−
1 −
88.88.88.88. ( ) ( )
5 3
2 4 2
1 1 2 2 2
5 3
t t
x x dx t t tdt t t dt C
 
− ⋅ = + ⋅ = + = + + = 
 
∫ ∫ ∫
2 2
1 1 1 2x t x t x t dx tdt− = ⇒ − = ⇒ = + ⇒ =
( ) ( )
5 3
5 3 2 1 2 12 2
5 3 5 3
x xt t
C C
− −
= + + = + +

25
89.89.89.89.
2 2
2 2 2 2 2 2 2
2
a sen t a a
a b x dx a b cost dt a a sen t cost dt
b b b
− ⋅ = − ⋅ ⋅ ⋅ = − ⋅ ⋅ ⋅ =∫ ∫ ∫
2 2 2 2 2
2
1
asent a
x dx costdt
b b
bx b x a b x
sent cost
a a a
bx
t arcsen
a
= ⇒ =
−
= ⇒ = − =
 
=  
 
( )2 2 2 2
1
a a
a sen t cost dt a cos t cost dt
b b
= − ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ =∫ ∫
2 2
2 1 2
2
a a a cos t
a cost cost dt cos t dt dt
b b b
+
= ⋅ ⋅ ⋅ ⋅ = ⋅ = =∫ ∫ ∫
2 2 2 2
2
2 4 2 2
a t a sen t a a sent cost
C t C
b b b b
⋅
= ⋅ + ⋅ + = ⋅ + ⋅ + =
2 2 2 2 2
2 2
a b a bx a b x
arcsen x C
b a b a a
− 
= + ⋅ ⋅ + = 
 
2
2 2 2
2 2
b
a arcsen x
x a b xa
C
b
 
⋅   − = + +
90.90.90.90.
2
1
1
x
x x
x
e
dx e ln e C
e
= − + +
+∫ (Es la misma que la nº 87)
91.91.91.91. ( )
4 3 2 4 3 2
3 2 2 3 2 3 1
2 3 7 7 7
4 3 2 4 3 2
x x x x x x
x x x lnx dx x lnx x
x
   
− + − ⋅ = − + − − − + −   
   
∫ ∫
( )
4 3 2
3 2
1
2 3
2 3 7 7
4 3 2
u lnx du dx
x
x x x
dv x x x dx v x
= ⇒ =
= − + − ⇒ = − + −
4 3 2 3 2
2 3 2 3
7 7
4 3 2 4 3 2
x x x x x x
x lnx dx
   
= − + − − − + − =   
   
∫
4 3 2 4 3 2
2 3 2 3
7 7
4 3 2 16 9 4
x x x x x x
x lnx x C
   
= − + − − − + − +   
   
92.92.92.92. ( )1 1 1 1
3 3 3 2 2
4
5 5 4 7
7
x dx x x dx
x
− − 
− = ⋅ − ⋅ ⋅ = 
 
∫ ∫

26
1 41 11 1
3 32 21 11 1
3 32 2
5 4 7 5 4 7
1 1 4 11 1
3 2 3 2
x x x x
C C
+ − +
− −
= ⋅ − ⋅ ⋅ + = ⋅ − ⋅ ⋅ + =
+ − +
433 33 8 3
5 5 8
4 4 77
x
x x C x x C− ⋅ + = − + =
3 33 8 7 3 8
5 5 7
4 7 4 7
x x x C x x x C− ⋅ + = − +

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